Powerful pp-groups have noninner automorphisms of order pp and some cohomology

In this paper we study the longstanding conjecture of whether there exists a noninner automorphism of order $p$ for a finite non-abelian $p$-group. We prove that if $G$ is a finite non-abelian $p$-group such that $G/Z(G)$ is powerful then $G$ has a noninner automorphism of order $p$ leaving either $\Phi(G)$ or $\Omega_1(Z(G))$ elementwise fixed. We also recall a connection between the conjecture and a cohomological problem and we give an alternative proof of the latter result for odd $p$, by showing that the Tate cohomology $H^n(G/N,Z(N))\not=0$ for all $n\geq 0$, where $G$ is a finite $p$-group, $p$ is odd, $G/Z(G)$ is $p$-central (i.e., elements of order $p$ are central) and $N\lhd G$ with $G/N$ non-cyclic.Comment: to appear in Journal of Algebr

Engel graph associated with a group

Let $G$ be a non-Engel group and let $L(G)$ be the set of all left Engel elements of $G$. Associate with $G$ a graph $\mathcal{E}_G$ as follows: Take $G\backslash L(G)$ as vertices of $\mathcal{E}_G$ and join two distinct vertices $x$ and $y$ whenever $[x,_k y]\not=1$ and $[y,_k x]\not=1$ for all positive integers $k$. We call $\mathcal{E}_G$, the Engel graph of $G$. In this paper we study the graph theoretical properties of $\mathcal{E}_G$.Comment: Proposition 2.8 is omitted however the proof is correct. Some errors and misprints are corrected. Corollary 2.11 (now Corollary 2.10) is now in a corrected for

Left 3-Engel elements in groups

In this paper we study left 3-Engel elements in groups. In particular, we prove that for any prime $p$ and any left 3-Engel element $x$ of finite $p$-power order in a group $G$, $x^p$ is in the Baer radical of $G$. Also it is proved that $$ is nilpotent of class 4 for every two left 3-Engel elements in a group $G$.Comment: A serious error in the proof Theorem 1.2, the correct proof, also correcting some misprint